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CSE 396 Fall2008 Introduction to the Theory of Computation Homework Solution - Set 1 Due: Friday 09/05/08 1. Textbook, Page 26, Exercise 0.7. Let A = {a, b, c}. Let Z+ be the set of positive integers. a. The following relation R1 is a binary relation on A that is reflexive and symmetric but not transitive: R1 = {(a, a), (b, b), (c, c), (a, b), (b, a), (b, c), (c, b)}. The following relation R2 is an infinite such relation, which is defined on Z+ : R2 = {(a, b) | ka − bk ≤ 1} b. The following relation R3 is a binary relation on A that is reflexive and transitive but not symmetric: R3 = {(a, a), (b, b), (c, c), (a, b), (b, c), (a, c)}. The normal ≤ relation defined on Z+ is an infinite such relation. c. The following relation R4 is a binary relation on A that is symmetric and transitive but not reflexive: R4 = {(a, a), (b, b), (a, b), (b, a)}. The following relation R5 is an infinite such relation, which is defined on Z+ : R2 = {(a, b) | both a and b are even. } Note that the key point in constructing a relation that is symmetric and transitive but not reflexive is that there has to be some “isolated” element in the base set that is not related to any other element. 1 2. Textbook, Page 27, Exercise 0.8. The graph is as follows: 1 4 2 3 The thickened lines indicate a path from node 3 to node 4. The degree of node 1 is 3 and that of node 3 is 2. 2